On the Number of Places of Convergence for Newton's Method over Number Fields

Details On the Number of Places of Convergence for Newton's Method over Number Fields On the Number of Places of Convergence for Newton's Method over Number Fields

2010-03-05

arxiv.org/abs/1003.1236v2

Mathematics

Number Theory

9 pages; minor changes from the previous version; to appear in Journal de Th\'eorie des Nombres de Bordeaux

Scientific paper

Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 be a sufficiently general element of K, and let alpha be a root of f. We give precise conditions under which Newton iteration, started at the point x_0, converges v-adically to the root alpha for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.

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